- ており is used in the middle of a sentence instead ている, but it's not the middle of a sentence here.

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Does this make sense to you?
部活前に、「これからシンガポール部」ということを呟いたら、部活中に、「シンガポール大好き！」「シンガポール料理大好き！」と、二人が盛り上がっており、そんな身近で、シンガポール好きが二人もいたとは、と驚き、

「じゃあ、今度3人で行きましょう！」ということになりました。

You shouldn't let his inter-punctuation trouble you... (。->、) It may have been a mistake, or intentional, but it doesn't really matter. Also, sometimes the 連用形 is used to "end" a sentence which produces an effect similar to ending an English sentence with an ellipsis (eg "And then he went...")

連用形 also loosely connects sentences, it is not uncommon to find Japanese long sentences comprised of (almost) independent smaller sentences loosely connected this way, which might as well be rewritten as many smaller sentences. In English, we can use "and" to connect sentences three A. B. C. loosely: "A and B and C." Compare the English:

“Elizabeth, New Jersey, when my mother was being raised there in a flat over her father’s grocery store, was an industrial port a quarter the size of Newark, dominated by the Irish working class and their politicians and the tightly knit parish life that revolved around the town’s many churches, and though I never heard her complain of having been pointedly ill-treated in Elizabeth as a girl, it was not until she married and moved to Newark’s new Jewish neighborhood that she discovered the confidence that led her to become first a PTA “grade mother,” then a PTA vice president in charge of establishing a Kindergarten Mothers’ Club, and finally the PTA president, who, after attending a conference in Trenton on infantile paralysis, proposed an annual March of Dimes dance on January 30 – President Roosevelt’s birthday – that was accepted by most schools.”

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New Jersey was an industrial port a quarter the size of Newark. My mother was being raised there in a flat over her father’s grocery store. [It was] dominated by the Irish working class and their politicians and the tightly knit parish life that revolved around the town’s many churches. I never heard her complain of having been pointedly ill-treated in Elizabeth as a girl. It was not until she married and moved to Newark’s new Jewish neighborhood that she discovered the confidence that led her to become first a PTA “grade mother,” then a PTA vice president in charge of establishing a Kindergarten Mothers’ Club. And finally the PTA president. After attending a conference in Trenton on infantile paralysis, [she] proposed an annual March of Dimes dance on January 30 (President Roosevelt’s birthday). [It] was accepted by most schools.

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When in doubt, try to understand the sentence pretending there were no periods/commas/... . In English, you could write something like "I want. To go. And eat." and everybody would understand...

Btw, in classical Japanese, おり would be the 終止形 of おる (similar to あり,なり), so 熊なり。would be a complete sentence, but this certainly doesn't apply here.

Is おり more formal, as you mentioned, a humble form of いて（いる）？ Yes because they are interchangeable:
部活中に、「シンガポール大好き！」「シンガポール料理大好き！」と、二人が盛り上がっていて。
そんな身近で・・
These still work and don't differ. I don't think I can find problems in your understanding.

For 連用形 of いる, おり is sometimes preferred to the very short い. NHK, the public broadcaster of Japan, use いて, which replaced おり some time ago (well, I don't remember exactly but about 15 years ago) in their usage manual. おり has a slightly formal tone but it is still used extensively in writing and in conversations for which いて sounds too casual.

Using "。" where "、" is appropriate as in this blog entry is permissible only as slang on the 'Net. It's not recommended for learners.